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330 XI ELEMENTARY SPECTRAL THEORY

Note that the proof that «*(B) is precompact still holds when E is not
complete; but it can happen that in a prehilbert space E, a compact operator
has an adjoint which is not compact.

(1 1 .5.5) Let u be a compact operator in a complex prehilbert space E, having
an adjoint u* which is compact. Then :

(a) The spectrum sp(w*) is the image of$p(u) by the mapping £ -* £ .

(b) For each X ^ 0 in sp(w), k(l; u) = k$\ «*).

(c) Ifv = u — A • 1_E, then v*(E) is the orthogonal supplement (Section 6.3)
of v ~ ¹ (0) = E(/l ; u), and the dimensions of the eigenspaces E(l ; u) and E(l ; w*)
are equal

(d) The subspace F(I; w*) is the orthogonal supplement o/N(A; u), and the
dimensions 0/N(A; u) and N(I; «*) a

We have v* = w* - I • 1_E , hence (t?(;c) | j) = (jc | v*(y)) from (1 1 .5.1 ), and
therefore the relation v(x) = 0 implies that x is orthogonal to the subspace
t>*(E). Now by (11.4.1) applied to w*, v*(E) is the topological direct sum
of F(I; w*) and of the subspace y*(N(I; u*)) of N(l; w*), and from linear
algebra (A.4.17) it follows that the codimension of z;*(E) is equal to the di-
mension of v*"¹^) = E(l; w*); hence we have dim E(A; w) < dim E(I; w*).
But M = (M*)*_? hence we have dim E(A; w) = dim E(l; w*); furthermore, the
orthogonal supplement of E(A; w) contains u*(E) and has the same Co-
dimension as u*(E), hence both are equal, which proves (c). This also shows
that for any eigenvalue 1 + 0 of w, I is an eigenvalue of M*, and as the con-
verse follows from the relation u = («*)*, we have also proved (a).

The same argument may be applied to the successive iterates v^h of v, and
shows that the image of E by v*^h = (v^h)* is the orthogonal supplement of the
kernel of v^h. Using (11.3.2), (11.4.1), and the relation u = (w*)*, this im-
mediately proves (b) and (d).

Theorems (11.4.1) and (11.5.5) can be translated into a criterion for the
solutions of the equation u(x) — kx = y:

(11.5.6) Under the assumptions of (11.5.5) ;

(a) If A, is not in the spectrum ofu, the equation u(x) — kx — y has a unique
solution in Efor every yeE.

(b) If A, ^ 0 is in the spectrum ofu, a necessary and sufficient condition for
y e E to be such that the equation u(x) — Ix = y have a solution in E is that y
be orthogonal to the solutions of the equation u*(x) — Ix = 0.



	330 XI ELEMENTARY SPECTRAL THEORY Note that the proof that «*(B) is precompact still holds when E is not complete; but it can happen that in a prehilbert space E, a compact operator has an adjoint which is not compact.

	(1 1 .5.5) Let u be a compact operator in a complex prehilbert space E, having an adjoint u* which is compact. Then : (a) The spectrum sp(w) is the image of$p(u) by the mapping £* -* £ . (b) For each X ^ 0 in sp(w), k(l; u) = k$\ «). (c) Ifv* = u — A • 1_E, then v(E) is the orthogonal supplement* (Section 6.3) of v ~ ¹ (0) = E(/l ; u), and the dimensions of the eigenspaces E(l ; u) and E(l ; w) are equal* (d) The subspace F(I; w) is the orthogonal supplement* o/N(A; u), and the dimensions 0/N(A; u) and N(I; «*) a

	We have v* = w* - I • 1_E , hence (t?(;c) \| j) = (jc \| v(y))* from (1 1 .5.1 ), and therefore the relation v(x) = 0 implies that x is orthogonal to the subspace t>(E). Now by (11.4.1) applied to w, v(E)* is the topological direct sum of F(I; w) and of the subspace y(N(I; u))* of N(l; w), and from linear algebra (A.4.17) it follows that the codimension* of z;(E) is equal to the di- mension of v"¹^) = E(l; w); hence we have dim E(A; w) < dim E(I; w). But M = (M)_? hence we have dim E(A; w) = dim E(l; w); furthermore, the orthogonal supplement of E(A; w) contains u(E) and has the same Co- dimension as u(E), hence both are equal, which proves (c). This also shows that for any eigenvalue 1 + 0 of w, I is an eigenvalue of M, and as the con- verse follows from the relation u = («), we have also proved (a). The same argument may be applied to the successive iterates v^h of v, and shows that the image of E by v^h = (v^h) is the orthogonal supplement of the kernel of v^h. Using (11.3.2), (11.4.1), and the relation u = (w), this im- mediately proves (b) and (d). Theorems (11.4.1) and (11.5.5) can be translated into a criterion for the solutions of the equation u(x) — kx = y:

	(11.5.6) Under the assumptions of (11.5.5) ; (a) If A, is not in the spectrum ofu, the equation u(x) — kx — y has a unique solution in Efor every yeE. (b) If A, ^ 0 is in the spectrum ofu, a necessary and sufficient condition for y e E to be such that the equation u(x) — Ix = y have a solution in E is that y be orthogonal to the solutions of the equation u*(x) — Ix = 0.